Properties

Degree $2$
Conductor $5$
Sign $-0.0454 - 0.998i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.79e4 − 3.79e4i)2-s + (3.04e7 − 3.04e7i)3-s − 1.41e9i·4-s + (7.42e10 − 1.33e11i)5-s − 2.31e12·6-s + (−9.26e12 − 9.26e12i)7-s + (−2.16e14 + 2.16e14i)8-s + 9.78e11i·9-s + (−7.88e15 + 2.24e15i)10-s + 1.98e16·11-s + (−4.29e16 − 4.29e16i)12-s + (−8.93e17 + 8.93e17i)13-s + 7.03e17i·14-s + (−1.79e18 − 6.31e18i)15-s + 1.03e19·16-s + (−3.66e19 − 3.66e19i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.579i)2-s + (0.706 − 0.706i)3-s − 0.328i·4-s + (0.486 − 0.873i)5-s − 0.819·6-s + (−0.278 − 0.278i)7-s + (−0.769 + 0.769i)8-s + 0.000527i·9-s + (−0.788 + 0.224i)10-s + 0.432·11-s + (−0.232 − 0.232i)12-s + (−1.34 + 1.34i)13-s + 0.323i·14-s + (−0.273 − 0.961i)15-s + 0.563·16-s + (−0.753 − 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0454 - 0.998i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0454 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.0454 - 0.998i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.0454 - 0.998i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.4696101500\)
\(L(\frac12)\) \(\approx\) \(0.4696101500\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.42e10 + 1.33e11i)T \)
good2 \( 1 + (3.79e4 + 3.79e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-3.04e7 + 3.04e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (9.26e12 + 9.26e12i)T + 1.10e27iT^{2} \)
11 \( 1 - 1.98e16T + 2.11e33T^{2} \)
13 \( 1 + (8.93e17 - 8.93e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (3.66e19 + 3.66e19i)T + 2.36e39iT^{2} \)
19 \( 1 + 2.80e20iT - 8.31e40T^{2} \)
23 \( 1 + (2.06e21 - 2.06e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.02e23iT - 6.26e46T^{2} \)
31 \( 1 + 9.64e23T + 5.29e47T^{2} \)
37 \( 1 + (-7.27e24 - 7.27e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 1.58e25T + 4.06e51T^{2} \)
43 \( 1 + (9.21e25 - 9.21e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (3.74e26 + 3.74e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-4.43e27 + 4.43e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 4.27e27iT - 4.64e56T^{2} \)
61 \( 1 + 5.42e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.45e29 - 1.45e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 1.17e29T + 1.73e59T^{2} \)
73 \( 1 + (8.02e28 - 8.02e28i)T - 4.22e59iT^{2} \)
79 \( 1 + 2.05e30iT - 5.29e60T^{2} \)
83 \( 1 + (4.06e30 - 4.06e30i)T - 2.57e61iT^{2} \)
89 \( 1 + 1.23e31iT - 2.40e62T^{2} \)
97 \( 1 + (-2.90e31 - 2.90e31i)T + 3.77e63iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.44378499498831624130993976672, −13.27859911620536849366686318460, −11.65432421536946885083130403043, −9.671210567413512533799918881743, −8.824253368084103420529460706288, −6.94436424055258988289756763325, −4.89401115117803779520774662122, −2.40009008444455147743097926826, −1.56897917611106422276447025307, −0.14750831446699077080064432824, 2.58262807734597938689910146040, 3.70976761500393408280920809757, 6.15035492681996206929515258446, 7.68779647052863845998996514924, 9.174360082856701450177340534683, 10.21730624924337132877049796568, 12.54988592898841257805497664619, 14.61225815535622945122503493555, 15.42260040369571149221334503298, 17.09789632801578234022045131066

Graph of the $Z$-function along the critical line